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<title>FFTW 3.3.8: The Halfcomplex-format DFT</title>

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<a name="The-Halfcomplex_002dformat-DFT"></a>
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Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="The-Halfcomplex_002dformat-DFT-1"></a>
<h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>

<p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
<a name="index-FFTW_005fR2HC"></a>
<a name="index-r2c-1"></a>
<a name="index-r2hc"></a>
(see <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>) but with &ldquo;halfcomplex&rdquo;
format output, and may sometimes be faster and/or more convenient than
the latter.
<a name="index-halfcomplex-format-1"></a>
The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
<a name="index-FFTW_005fHC2R"></a>
<a name="index-hc2r"></a>
This consists of the non-redundant half of the complex output for a 1d
real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
numbers (<code>double</code>) in the format:
</p>
<p align=center>
r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
</p>

<p>Here,
r<sub>k</sub>
is the real part of the <em>k</em>th output, and
i<sub>k</sub>
is the imaginary part.  (Division by 2 is rounded down.) For a
halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
zero due to symmetries of the real-input DFT, and is not stored.
Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
length <code>n</code>, and vice versa for hc2r.
<a name="index-normalization-2"></a>
</p>

<p>Aside from the differing format, the output of
<code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
the corresponding 1d r2c/c2r transform
(i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
Recall that these transforms are unnormalized, so r2hc followed by hc2r
will result in the original data multiplied by <code>n</code>.  Furthermore,
like the c2r transform, an out-of-place hc2r transform will
<em>destroy its input</em> array.
</p>
<p>Although these halfcomplex transforms can be used with the
multi-dimensional r2r interface, the interpretation of such a separable
product of transforms along each dimension is problematic.  For example,
consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
FFTW_R2HC, FFTW_MEASURE)</code>.  Conceptually, FFTW first transforms the rows
(of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
columns (of size <code>n0</code>).  Half of these column transforms, however,
are of imaginary parts, and should therefore be multiplied by <em>i</em>
and combined with the r2hc transforms of the real columns to produce the
2d DFT amplitudes; FFTW&rsquo;s r2r transform does <em>not</em> perform this
combination for you.  Thus, if a multi-dimensional real-input/output DFT
is required, we recommend using the ordinary r2c/c2r
interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
</p>
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Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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